Re: How many real numbers are there?

Mike Deeth wrote:
Eric Schmidt wrote:

(snip)

You have not proved that every set (or even every real number) is in
some S_i. In fact, you yourself construct a set not in any S_i, namely
T. Therefore you have not found any flaw in set theory.



Of course my T does not contain every set, that would be really silly.
However, the elements of my set T satisfy the ZF axioms. They do this
by very construction, and you cannot argue about that! For example, if
x and y are in my set T, then so is x union y, {x,y}, etc.

The construction of the real numbers does not depend on the specific
details of the set theory. *** All it requires is that ZF axioms hold,
and they DO hold for elements of my set T *** Therefore, we can
construct the "reals" (ie, a complete ordered field) using nothing but
the elements of my set T. My set T has countably many elements.
THEREFORE the cardinality of the reals is countable!!

If you want to dispute this, please consider the following string of
statements and tell me which one is false. Just one will suffice.

1. The construction of the reals (ie, a complete ordered field) does
not depend on particular details of one's set theory, only on the fact
that the ZF axioms hold

2. ZF axioms hold for elements of my set T

3. We can construct, in the usual way, the "reals" (ie, a complete
ordered field), in such a way that each "real" is an element of my set
T

4. My set T has countable cardinality

5. THEREFORE, the "reals" (ie, one particular complete ordered field,
namely the one in statement 3) have countable cardinality.


Note that in making the deduction in step 5 we use the lemma "A subset
of a countable set is countable". As some point out, this might take a
little more power than ZF. But this is fine! My set T was constructed
in ZFC! The *elements* of my set T may or may not satisfy a choice
axiom. The complete ordered field constructed in statement 3 consists
of elements of T, and each element of T is a set in ZFC, and therefore
in the original ZFC I started out with, the complete ordered field of
statement 3 is a subset of the countable set T, and thus is countable.

The *very most* the evil Cantorians can salvage at this point would be
a statement like, "the assertion that the reals are uncountable, like
the axiom of choice or continuum hypothesis, is an independent
assertion and can't be proved or disproved".

I hope you're having a wonderful day,
Nathan the Great
Age 11 :-)


I think your construction can also be used to show that there are only countably many sets - you don't have to limit yourself to real numbers here. I think that you are trying to get to a proof that ZF or ZFC has a countable model.

Or to put it another way, I think your argument fails at point (3), because you can only show that T is complete in the sense that every constructable subset of T has a least upper bound in T - you cannot show this for every subset of T.

Stephen
.

Relevant Pages

  • Re: Irrelevant Relative Cardinality
    ... >>> the axioms of an ordered field, which you should know and which do not ... >> reals from the rationals, to show that there actually exists something ... rationals that does not in any way depend on Dedekind cuts. ...
    (sci.math)
  • Re: are Real Numbers evil?
    ... I am not honestly that familiar with how you use them for rotations ... that going around a triangle or a tetrahedron ... I do not know of any other natural construction that can do this. ... Reals are two-signed. ...
    (comp.theory)
  • Re: The real numbers, and general comments
    ... But if you want to talk about the uncountability of the reals, ... should know that the construction of the reals (whether by Dedekind cuts ... ZF is a set of axioms for set theory. ...
    (sci.math)
  • Re: are Real Numbers evil?
    ... that essentially even if understanding ... that going around a triangle or a tetrahedron ... > I do not know of any other natural construction that can do this. ... Reals are two-signed. ...
    (comp.theory)
  • Re: Possible flaw in the polynomial ring A[X] construction
    ... The less cryptic the construction then the better it will ... polynomial algebra is already inherent within the ring definition. ... approaching any tuple notation or definitions. ... being in the set of reals and all being the value 2.0. ...
    (sci.math)